Archive for July, 2017

Last time we determined the value for one of the key variables in the Euler-Eytelwein Formula known as the angle of wrap. To do so we worked with the relationship between the two tensions present in our example pulley-belt assembly, T1 and T2. Today we’ll use physics to solve for T2 and arrive at theMechanical Power Formula, which enables us to compute the amount of power present in our pulley and belt assembly, a common engineering task.

To start things off let’s reintroduce the equation which defines the relationship between our two tensions, the Euler-Eytelwein Formula, with the value for e, Euler’s Number, and its accompanying coefficients, as determined from our last blog,

T1 = 2.38T2 (1)

Before we can calculate T1 we must calculate T2. But before we can do that we need to discuss the concept of power.

The Mechanical Power Formula in Pulley and Belt Assemblies

Generally speaking, power, P, is equal to work, W, performed per unit of time, t, and can be defined mathematically as,

P = W ÷ t (2)

Now let’s make equation (2) specific to our situation by converting terms into those which apply to a pulley and belt assembly. As we discussed in a past blog, work is equal to force, F, applied over a distance, d. Looking at things that way equation (2) becomes,

P = F × d ÷ t (3)

In equation (3) distance divided by time, or “d ÷ t,” equals velocity, V. Velocity is the distance traveled in a given time period, and this fact is directly applicable to our example, which happens to be measured in units of feet per second. Using these facts equation (3) becomes,

P = F × V (4)

Equation (4) contains variables that will enable us to determine the amount of mechanical power, P, being transmitted in our pulley and belt assembly.

The force, F, is what does the work of transmitting mechanical power from the driving pulley, pulley 2, to the passive driven pulley, pulley1. The belt portion passing through pulley 1 is loose but then tightens as it moves through pulley 2. The force, F, is the difference between the belt’s tight side tension, T1, and loose side tension, T2. Which brings us to our next equation, put in terms of these two tensions,

P = (T1 – T2) × V (5)

Equation (5) is known as the Mechanical Power Formulainpulley and belt assemblies.

The variable V, is the velocity of the belt as it moves across the face of pulley 2, and it’s computed by yet another formula. We’ll pick up with that issue next time.

Sometimes things which appear simple turn out to be rather complex. Such is the case with the Euler-Eytelwein Formula, a small formula with a big job. It computes how friction, an omnipresent phenomenon in mechanical assemblies, contributes to the transmission of mechanical power. Today we’ll determine the value of one of the Euler-Eytelwein Formula’s variables, the angle of wrap.

The formula introduced last time to calculate the angle of wrap, θ, is,

θ = (180 – 2α) × (π ÷180) (2)

where,

α = sin-1((D1 – D2) ÷2x) (3)

By direct measurement we’ve determined the pulleys’ diameters, D1 and D2, are equal to 1 foot and 0.25 feet respectively. The term x is the distance between the two pulley shafts, 3 feet. The term sin-1 is a trigonometric function known as inverse sine, a button commonly found on scientific calculators.

Inserting our known values into equation (3) we arrive at,

α = sin-1((1.0 foot – 0.25 feet) ÷2 ×(3 feet)) (4)

α = 7.18 (5)

We can now incorporate equation (5) into equation (2) to solve for θ,

θ = (180 – (2 × 7.18)) ×(π ÷180) (6)

θ = 2.89 (7)

Inserting the values for m and θ into equation (1) we arrive at,

T1 = T2 × 2.718(0.3 × 2.89) (8)

T1 = 2.38T2 (9)

We have at this point solved for over half of the unknown variables in the Euler-Eytelwein Formula. We still can’t solve for T1, because we don’t know the value of T2. But that will change next time when we introduce yet another formula, this one to determine the amount of mechanical power present in our pulley-belt system.

Last time we introduced a scenario involving a hydroponics plant powered by a gas engine and multiple pulleys. Connecting the pulleys is a flat leather belt. Today we’ll take a step further towards determining what width that belt needs to be to maximize power transmission efficiency. We’ll begin by revisiting the two T’s of the Euler-Eytelwein Formulaand introducing a formula to determine a key variable, angle of wrap.

The Angle of Wrap Formula

We must start by calculating T1, the tight side tension of the belt, which is the maximum tension the belt is subjected to. We can then calculate the width of the belt using the manufacturer’s specified safe working tension of 300 pounds per inch as a guide. But first we’ll need to calculate some key variables in the Euler-Eytelwein Formula, which is presented here again,

T1 = T2× e(μθ) (1)

We determined last time that the coefficient of friction, μ, between the two interfacing materials of the belt and pulley are, respectively, leather and cast iron, which results in a factor of 0.3.

The other factor shown as a exponent of e is the angle of wrap, θ, and is calculated by the formula,

θ = (180 – 2α) ×(π ÷180) (2)

You’ll note that this formula contains some unique terms of its own, one of which is familiar, namely π, the other, α, which is less familiar. The unnamed variable α is used as shorthand notation in equation (2), to make it shorter and more manageable. It has no particular significance other than the fact that it is equal to,

α = sin-1((D1 – D2) ÷2x) (3)

If we didn’t use this shorthand notation for α, equation (2) would be written as,

θ = (180 – 2(sin-1((D1 – D2) ÷2x))) ×(π ÷180) (3a)

That’s a lot of parentheses!

Next week we’ll get into some trigonometry when we discuss the diameters of the pulleys, which will allow us to solve for the angle of wrap.